System Identification, Estimation, and Prediction of Advertising-Related Data

ABSTRACT

In accordance with the invention, a system, method, and apparatus for analyzing advertisement-related data are presented, which may include receiving data related to an aspect of an advertisement and modeling the aspect of the advertisement with a mathematical model. The mathematical model may include a control-signal-related component, a control-signal-independent component, and an error component. Each component may be updated based on at least one of a control signal, the received data, and a previous state of at least one of the components. An updated model may be created based on the updated components. The system, method, and apparatus may also include predicting the aspect of the advertisement using the updated model. Exemplary aspects of and data related to the advertisement may include one or more of the following: a number of impressions, “clicks,” or “conversions” and/or the impression-to-conversion, impression-to-click, or click-to-conversion ratios.

FIELD OF THE INVENTION

This invention is generally related to advertising. This invention ismore specifically related to system identification, estimation, andprediction of advertising-related data.

BACKGROUND OF THE INVENTION

Since the early 1990's, the number of people using the World Wide Webhas grown at a substantial rate. As more users take advantage of theWorld Wide Web, they generate higher and higher volumes of traffic overthe Internet. As the benefits of commercializing the Internet can betremendous, businesses increasingly take advantage of this traffic byadvertising their products or services on-line. These advertisements mayappear in the form of leased advertising space (e.g., “banners”) onwebsites or advertisements on digital television, which are comparableto rented billboard space in highways and cities or commercialsbroadcasted during television or radio programs.

Before users browse to a particular web page, there is much unknown tothe companies that might advertise on that page and to advertisers thatmay place advertisements on the companies' behalf. Neither the companiesnor the advertisers know how many users will browse to particular webpages and may or may not know of the number of competing advertisersthat are interested in advertising on the same web page, and thereforedo not know the volume of advertisements (the number of “impressions”)they will be able to place. Further, neither know how many users willselect or “click” on each advertisement or if “conversions,” e.g., salesor signing up new users, will result from each display or impression ofan advertisement or what the ratio of clicks to conversions may be.

Companies may be interested in impressions (e.g., if they are trying toincrease awareness of a brand), clicks (e.g., if they are trying toprovide more information about a product), or conversions (e.g., if theyare trying to make sales or get new users to sign up for services,etc.). Companies may pay, on the other hand, based on impressions,clicks, or conversions, or a combination thereof, regardless of theirinterests. In addition to wanting to predict impressions, clicks, andconversions, one may want to predict other data related to theadvertisement, such as the ratio of impressions to clicks or conversionsor the ratio of clicks to conversions. Moreover, one may want toidentify the relationship between control signals and advertisementrelated data.

It is accordingly an object of the invention to provide such predictionsand other advertising-related data.

SUMMARY OF THE INVENTION

In accordance with the invention, a system, method, and apparatus forprocessing advertisement-related data are presented, which may includereceiving data related to an aspect of an advertisement; modeling theaspect of the advertisement with a mathematical model, where themathematical model includes a control-signal-related component, acontrol-signal-independent component, and an error component; updatingeach component of the mathematical model based on at least one of aparticular control signal, the received data, and a previous state of atleast one of the components to create an updated model; and predictingthe aspect of the advertisement using the updated model. Someembodiments may further include the step of receiving one or morecontrol signals related to the received data, where the one or morecontrol signals include the particular control signal.

In some embodiments, the error component may be a multiplicative errorcomponent. The received data may include a number of times that theadvertisement has been shown; a number of times that the advertisementhas been selected by a user; a number of times that a conversion hasoccurred based on the advertisement; a probability that an impressionwill result in a selection; a probability that an impression will resultin a conversion; or a probability that a selection by a user will resultin a conversion.

The conversion may be a sale of an item, filling out a form, taking asurvey, and watching a sequence of web pages. The probability that aselection by a user will result in a conversion may be a probabilitythat selection by the user will result in a sale.

An advertisement may be an internet based advertisement, a digitaltelevision advertisement; an advertisement on a cell phone, or anadvertisement on a personal digital assistant. The advertisement mayalso be one of a plurality of advertisements and one may choose whichadvertisement among the plurality of advertisements to display based atleast in part on the prediction of the aspect of the advertisement. Thechoice of which advertisement to display may further be based on thecontrol signal.

The mathematical model may be a log-linear model of the form

g(u(k))^(Ω_(l_(u), m_(u))^(u)(k))^(Ω_(l, m)(k) + Cx_(ɛ)(k)),

a logit-linear model of the form

$\frac{{g( {u(k)} )}^{\Omega_{l_{u},m_{u}}^{u}{(k)}}^{{\Omega_{l,m}{(k)}} + {ɛ{(k)}} + {v{(k)}}}}{1 + {{g( {u(k)} )}^{\Omega_{l_{u},m_{u}}^{u}{(k)}}^{{\Omega_{l,m}{(k)}} + {ɛ{(k)}} + {v{(k)}}}}},$

or any other appropriate model.

The mathematical model may be updated using an adaptive estimationscheme, a standard Kalman filter, an extended Kalman filter, or anunscented Kalman filter, or any other appropriate technique.

The error component may include an autoregressive moving averageprocess. The control-signal-related component and thecontrol-signal-independent component may include periodic functions,which may be sums of one or more sine or cosine functions of periodsthat are multiples of 24 hours.

The aspect may be a number of times that the advertisement is shown, anumber of times that the advertisement is selected by a user, a numberof times that a conversion occurs based on the advertisement, aprobability that an impression results in selection by a user, aprobability that an impression results in a sale; or a probability thata selection by a user results in a conversion.

Additional objects and advantages of the invention will be set forth inpart in the description which follows, and in part will be obvious fromthe description, or may be learned by practice of the invention. Theobjects and advantages of the invention will be realized and attained bymeans of the elements and combinations particularly pointed out in theappended claims.

It is to be understood that both the foregoing general description andthe following detailed description are exemplary and explanatory onlyand are not restrictive of the invention, as claimed.

BRIEF DESCRIPTION OF THE DRAWINGS

The accompanying drawings, which are incorporated in and constitute apart of this specification, illustrate embodiments of the invention andtogether with the description, serve to explain the principles of theinvention. In the drawings:

FIG. 1 is a flow chart that depicts the allocation of advertising space,consistent with embodiments of the present invention.

FIG. 2 is a diagram that depicts the allocation of advertising space,consistent with embodiments of the present invention.

FIG. 3 is a block diagram depicting an apparatus and system forpredicting and controlling advertising-related data, consistent withembodiments of the present invention.

FIG. 4 is a flowchart depicting a method for predicting and estimatingaspects of advertising-related data, consistent with embodiments of thepresent invention.

FIG. 5 is a flow chart of an exemplary method for updating amathematical model of the aspect, consistent with embodiments of thepresent invention.

FIG. 6 is a chart of example impression volume data over time,consistent with embodiments of the present invention.

DESCRIPTION OF THE EMBODIMENTS

Reference will now be made in detail to exemplary embodiments of theinvention, examples of which are illustrated in the accompanyingdrawings. Wherever possible, the same reference numbers will be usedthroughout the drawings to refer to the same or like parts.

FIG. 1 is a flow chart that depicts the allocation of advertising space,consistent with certain embodiments of the present invention. Companiesmay wish to place advertisements on the Internet, on digital television,cell phones, personal digital assistants, or any other appropriatedevice. As such, they may pay for the services of advertisers, who mayplace ads on their behalf. FIG. 2 is a diagram that depicts theallocation of advertising space, consistent with embodiments of thepresent invention. A particular advertiser (an advertising company thatplaces advertisements on behalf of sponsoring companies) may havemultiple advertisements 260A, 260B, and 260C that it could place inadvertising space 270 on web page 280. Similarly, advertising space 270could be available as part of a digital broadcast 280 of a digitaltelevision signal. The advertiser may, as depicted in step 110, predictone or more aspects (such as viewings of the advertisement, revenue,etc.) for two or more advertisements 260A, 260B, and 260C. In someembodiments, the advertiser may receive historical data for eachadvertisement 260A, 260B, and 260C, model aspects of the advertisement,update the models whenever new data related to the advertisement isreceived, and predict aspects of the advertisement, such as revenue,based on the models.

Once the revenue and/or other aspects for the advertisements arepredicted in step 110, the advertiser may chose how to allocateadvertising space 270 among advertisements 260A, 260B, or 260C in step120. When determining the allocation of available advertising spaceamong candidate advertisements, one may estimate certain aspects ofadvertising-related data. For example, one may want to predict orestimate volume information such as number of impressions, clicks, orconversions per time period. One may also want to predict or estimateperformance information or success rates, such as impression-to-clickratio, impression-to-conversion ratio, or click-to-conversion ratio. Thevolume and/or performance information may be useful for determining howto allocate available advertising space among candidate advertisements.The choice of how to allocate advertising space 270 among advertisements260A, 260B, or 260C may be based upon revenue, available budgets of thecompanies sponsoring advertisements 260A, 260B, and 260C, likelihood ofan advertisement 260A, 260B, or 260C being selected by a user or“clicked,” likelihood of an advertisement resulting in a successful“conversion” (e.g., making a sale or signing up a new user for services,etc.), a combination of those factors, or other appropriate factors.Step 120 may be performed before allocating each available advertisingspace 270 or may be performed once per time period (e.g., hourly ordaily).

In step 130, advertising space is allocated in the manner determined instep 120. If steps 110 and 120 are performed before allocating eachavailable advertising space 270, then, in step 130, the advertisement260A, 260B, or 260C with the highest predicted revenue (or conversionrate, etc.) may be placed in available advertising space 270. If steps110 and 120 are performed once during each time period, then thecandidate advertisements 260A, 260B, and 260C may be distributed amongavailable advertising spaces 270 during the time period based on theallocation determined in step 120. For example, if steps 110 and 120 areperformed hourly, then the allocation determined in step 120 mayindicate that twenty percent of available advertising spaces 270 duringthe time period should be filled by advertisement 260A, twenty-fivepercent by advertisement 260B, and the remaining fifty-five percent byadvertisement 260C.

In some embodiments, there may be additional constraints on theallocation of available advertising space in step 130. For example,there may be a cap on the placement of advertisements 260A, 260B, and260C during a particular time period based on the sponsoring company'srequirements, such as desired distribution of advertisements over timeor available budget. For example, a sponsoring company may not wish tospend over a certain amount on advertisements during the course of anadvertising campaign or may wish to conform the placements of theadvertisement to a particular temporal distribution, such as showing theadvertisement at most 100 times per hour. The allocation of availableadvertising space 270 may also be dependent on control signals u foreach advertisement 260A, 260B, and 260C.

FIG. 3 is a block diagram depicting an apparatus and system forpredicting and controlling advertising-related data, consistent withembodiments of the present invention. As a preliminary matter, a plantmay be a physical or logical system that takes one or more data streamsas input and outputs one or more data streams. The plant may be modeledwith a parametric mathematical model defined by a parameter vector x.For example, x may be parameters for a model that describes impressions,clicks, or conversions per time period; or impression-to-click ratio,impression-to-conversion ratio, or click-to-conversion ratio. A plantmay take zero, one or more control signals u as input and output actualrevenue, click volume, impression volume, etc. based on the placement ofan Internet advertisement on a series of web pages. Plant estimationmodule 310 may receive as input zero, one or more control signals u fromcontrol module 320 and volume and/or performance information from plantmodule 330. As described more herein, plant estimation module 310 mayuse the received information to produce {circumflex over (x)}, which isan estimate of the parameter vector x

In some embodiments, there may be more than one plant estimation module310 and each plant estimation module 310 may estimate a single set ofparameters {circumflex over (x)} for a single plant model. Therefore,there may be one plant estimation module 310 per volume or performancevariable. Numerous plant estimation modules 310 may work in concert tohelp control an advertising system.

Plant estimation module 310 may receive as input the volume and/orperformance signals that correspond to the plant it is attempting toestimate or model. For example, if plant estimation module 310 isestimating click-to-conversion ratio, then plant estimation module 310may receive a click-to-conversion ratio signal from plant module 330 andmay output parameters {circumflex over (x)} related to theclick-to-conversion ratio to control module 320. In some embodiments,plant estimation module 310 may additionally receive other signals. Forexample, if plant estimation module 310 is estimatingclick-to-conversion ratio, then it may receive click volume informationand conversion volume information from plant module 330. In someembodiments, plant estimation module 310 may use the received signals aswell as the control signals u to estimate parameters x.

Plant module 330 may take as input the control signal u from controlmodule 320 and output the result of the plant, such as the signalcorresponding to parameters {circumflex over (x)}. For example, if theplant is the click volume of a particular advertisement, then the plantmodule 330 may take as input a control signal u and attempt to controlthe click volume based on u. If there are multiple advertisements 260A,260B, and 260C, for example, and the plant corresponds to the clickvolume of advertisement 260A, then the plant module 330 may attempt toincrease or decrease the click volume (e.g., if u were increased ordecreased, respectively) by increasing or decreasing the allocation ofavailable advertising spaces 270 to advertisement 260A. If the plant isa performance ratio, such as impression-to-conversion ratio, then theplant module may attempt to increase the impression-to-conversion ratioby increasing the number of impressions that are shown on websites forwhich impressions are more likely to result in conversion and/ordecrease the number of impressions shown on websites that are lesslikely to result in conversion. The opposite balance may be struck todecrease a performance ratio.

Plant module 330 may directly allocate available advertising spaces 270when they become available or may be coupled to another device, module,or apparatus such as an ad server (not pictured) that allocatesavailable advertising spaces 270 as they become available.

After available advertising space 270 has been allocated and informationrelated to the relevant signal for plant module 330 becomes available(e.g., it may be hours or days after an advertisement 260A is displayedbefore conversion volume for advertisement 260A is available), therelevant signal is sent to plant estimation module 310. Plant estimationmodule 310 may use the relevant signal information to update one or moreestimation models related to the relevant signal and send the updatedparameter estimate {circumflex over (x)} for the models to controlmodule 320. Control module 320 may then alter control signal u in orderto obtain the desired allocation for the related advertisement. Forexample, if control module 320 receives the updated parameter estimate{circumflex over (x)} which model click volume for advertisement 260A,then control module 320 may alter the control signal u sent to plantmodule 330 in order to increase or decrease the click volume ofadvertisement 260A.

In certain embodiments, plant estimation module 310, control module 320and plant module 330 may be combined or they may each be coupled to theother two modules. Each of plant estimation module 310, control module320, and plant module 330 may also be coupled to one or more otherdevices, modules, or apparatuses (not pictured). The coupling discussedherein may include, but is not limited to, electronic connections,coaxial cables, copper wire, and fiber optics, including the wires thatcomprise a data bus. The coupling may also take the form of acoustic orlight waves, such as lasers and those generated during radio-wave andinfra-red data communications. Coupling may be accomplished bycommunicating control information or data through one or more networksto other data devices. In some embodiments, communication via couplingmay be accomplished by sending information directly from one device,module, or apparatus to another device, module or apparatus. In otherembodiments, communication may be accomplished via intermediate storagesuch as databases, flat files, binary files, registers, or one or morememories.

Each of the logical or functional modules described herein may comprisemultiple modules. The modules may be implemented individually or theirfunctions may be combined with the functions of other modules. Further,each of the modules may be implemented on individual components, or themodules may be implemented as a combination of components. For example,plant estimation module 310, control module 320, and plant module 330may each be implemented by a field-programmable gate array (FPGA), anapplication-specific integrated circuit (ASIC), a complex programmablelogic device (CPLD), a printed circuit board (PCB), a combination ofprogrammable logic components and programmable interconnects, a singleCPU chip, a CPU chip combined on a motherboard, a general purposecomputer, or any other combination of devices or apparatuses capable ofperforming the tasks of modules 310, 320, or 330. In some embodiments,plant estimation module 310, control module 320, and plant module 330may also include one or more memories that comprise a random accessmemory (RAM), a read only memory (ROM), a programmable read-only memory(PROM), a field programmable read-only memory (FPROM), or other dynamicstorage device, coupled to a data bus for storing information andinstructions to be executed by the module.

FIG. 4 is a flowchart depicting a method for predicting and estimatingaspects of advertising-related data, consistent with embodiments of thepresent invention. In step 410, data related to an advertisement isreceived by, for example, plant estimation module 310. In step 420, thereceived data is used to update a model of an aspect of theadvertisement, such as volume or performance. The updated model is thenused by, for example, control module 320 in step 430 to predict theaspect of the updated model. For example, if one were interested incontrolling the impression volume of an advertisement, then the datareceived in step 410 may include data related to impression volume andthe value of a scalar control signal u over time for the advertisement.In step 420, the impression volume may be used (along with, e.g., anextended or unscented Kalman filter) to update a model of the impressionvolume, including how control signal u may affect impression volume.Then, in step 430, the updated model may be used to predict futureimpression volume. Furthermore, the predictions may be used to predictwhat scalar value to use for control signal u_(desired) in order to geta desired impression volume (not pictured) and this control signalu_(desired) may then be sent, e.g., from control module 320 to plantmodule 330, to attempt to control the plant such that it produces thedesired impression volume.

In step 410, data related to an advertisement is received, and in step420 that data is used to update a model of an aspect of theadvertisement. The data received in step 410 may be related to anyaspect of the advertisement. The data may include, for example, controlsignals, volume data for impressions, clicks, or conversions. The datamay include performance ratios such as impression-to-click ratio,impression-to-conversion ratio, or click-to-conversion ratio. In someembodiments, any data that may be used to predict the aspect may bereceived in step 410. For example, if the aspect being modeled andupdated in step 420 is impression volume, then the data received in step410 may include impression data. If the aspect being modeled and updatedin step 420 is conversion volume, then the data received in step 410 mayinclude conversion volume, impression volume, and/or click volume. Insome embodiments, for example, impression-to-conversion ratio andimpression volume may be useful for helping predict conversion volume.

The data received in step 410 is used in step 420 to update a model ofan aspect of the advertisement. The aspect modeled may be a volumemeasure (e.g., impressions, clicks, or conversions) or a performancemeasure (e.g., impression-to-click ratio, impression-to-conversionratio, or click-to-conversion ratio). Volume and performance measuresmay comprise time-of-day and day-of-week features (Consider FIG. 6,which is a chart of example impression volume data over time, consistentwith embodiments of the present invention. Each data point may representthe number of impressions for an advertisement in a given hour.) Assuch, the model may comprise a control-signal-independent component thathas a truncated Fourier series, where/may be the number of terms in a24-hour periodic component and m may be the number of terms in a168-hour periodic component, and k may be the time period:

${{\Omega_{l,m}(k)} = {\beta_{0} + {\sum\limits_{i = 1}^{l}\; {\beta_{{24\; {hr}},i}{\sin ( {\frac{2\pi \; {ik}}{24} + \phi_{{24\; {hr}},i}} )}}} + {\sum\limits_{i = 1}^{m}\; {\beta_{{168\; {hr}},i}{\sin ( {\frac{2\pi \; {ik}}{168} + \phi_{{168\; {hr}},i}} )}}}}},$

where the 1+2(l+m) gain and phase parameters (the β's and φ's) definethe control-signal-independent component of the model. The parametersmay be represented by the following vector:

x₀=[β₀β_(24 hr,1),φ_(24 hr,1) . . .β_(24 hr,1)φ_(24 hr,1)β_(168 hr,1)φ_(168 hr,1) . . .β_(168 hr,m)φ_(168 hr,m)]^(T)

In other embodiments, components that are periodic to other than 24 and168-hours may be used. Further, components that incorporate mathematicalformulas or functions other than sine and cosine may be used.

Ω_(l,m)(k) and x₀ may correspond to plant behavior that is unrelated tothe control signal u(k) (control-signal-independent). There may also bea control-signal-related component, Ω_(l) _(u) _(,m) _(u) ^(u)(k).Control signal u(k) may affect the plant with a time-of-day and/orday-of-week periodicity. This may be modeled using a truncated Fourierseries defined in a similar manner as Ω_(l,m)(k). Thecontrol-signal-related component may be defined as follows:

${\Omega_{l_{u},m_{u}}^{u}(k)} = {\gamma_{0} + {\sum\limits_{i = 1}^{l_{u}}\; {\gamma_{{24\; {hr}},i}{\sin ( {\frac{2\pi \; {ik}}{24} + \phi_{{24\; {hr}},i}} )}}} + {\sum\limits_{i = 1}^{m_{u}}\; {\gamma_{{168\; {hr}},i}{{\sin ( {\frac{2\pi \; {ik}}{168} + \phi_{{168\; {hr}},i}} )}.}}}}$

The control-signal-related component may comprise 1+2(l_(u)+m_(u)) gainand phase parameters (the γ's and φ's), which may be represented by thefollowing vector:

x_(u)[γ₀γ_(24 hr,1)φ_(24 hr,1) . . .γ_(24 hr,1)φ_(24 hr,1)γ_(168 hr,1)φ_(168 hr,1) . . .γ_(168 hr,m)φ_(68 hr,m)]^(T)

In other embodiments, components that are periodic to other than 24 and168-hours may be used. Further, components that incorporate mathematicalformulas or functions other than sine and cosine may be used.

Although Ω_(l,m)(k) and Ω_(l) _(u) _(,m) _(u) ^(u)(k) may bestructurally similar, they may differ in the number of terms in the sumsl, l_(u), m, and m_(u) and in the values of x₀ and x_(u). In someembodiments, one or both of the control-signal-related component and thecontrol-signal independent component may have no daily or weeklyperiodicity. In such embodiments, the corresponding parameter (l, m,l_(u), and/or m_(u)) may be assigned the value zero. For example, if itis determined that the impact of control on the plant behavior isindependent of the time-of-day and day-of-week, l_(u) and m_(u) may beset to zero and Ω_(0,0) ^(u)(k) may in some embodiments be defined asfollows:

Ω_(0,0) ^(u)(k)=γ₀.

Combining the control-signal-independent and control-signal-relatedcomponents' parameters, one may have a vector x that represents a modelof the aspect:

${x = \begin{bmatrix}x_{0} \\x_{u}\end{bmatrix}},$

where x is a 2(1+l+m+l_(u)+m_(u))-dimensional column vector.

The model may also have an error component ε(k). ε(k) may be representedas a single parameter in x or it may comprise numerous parameters, eachof which may be included in x. For example, in some embodiments, ε(k)may be represented by an Auto-Regressive Moving Average (ARMA) process.ε(k) may be approximated by an ARMA (p, q) process

ε(k)=a ₁ε(k−1)+ . . . +a _(p)ε(k−p)+μ(k)+b ₁μ(k−1)+ . . . +b₁μ(k−q),  (1)

where μ(k) are samples from a white noise process of mean 0 and varianceσ_(μ) ² where

${E\lbrack {{\mu (k)}{\mu (l)}} \rbrack} = \{ \begin{matrix}{\sigma_{\mu}^{2},} & {k = l} \\{0,} & {k \neq l}\end{matrix} $

In Equation (1), ε(k) is represented by a one-dimensional differenceequation of order p. It may also be represented by a p-dimensionaldifference equation of order one (a state-space representation). Forexample, if

s=max(p,q+1),

a_(j)=0, for j>p,

b_(j)=0, for j>q, and

a₀=0

it may follow that

x_(ɛ)(k + 1) = Ax_(ɛ)(k) + B μ(k + 1) ɛ(k) = Cx_(ɛ)(k) where${A = {\begin{bmatrix}0 & 1 & 0 & \; & 0 \\0 & 0 & 1 & \; & 0 \\\vdots & \vdots & \vdots & \ddots & \vdots \\0 & 0 & 0 & \; & 1 \\a_{s} & a_{s - 1} & a_{s - 2} & \; & a_{1}\end{bmatrix} \in}},{B = {\begin{bmatrix}0 \\0 \\\vdots \\0 \\1\end{bmatrix} \in}},{and}$ C = [b_(s − 1)  b_(s − 2)  ⋯  b₀] ∈ .

where μ(k) may be white noise of mean zero and variance σ_(μ) ², andwhere A, B, and C may be constant matrices with dimensions Aε

, Bε

, and Cε

.

One may use the following log-linear model for volume n(k), where thevolume may relate to impressions, clicks, or conversions:

log(n(k))=log(g(u(k)))Ω_(l) _(u) _(,m) _(u) ^(u)(k)+Ω_(l,m)(k)+ε(k)  (2)

where ε(k)˜ARMA(p,q).

A plant system for a volume variable may be described with the followingstate-space equations:

$\begin{matrix}{{{x_{ɛ}( {k + 1} )} = {{{Ax}_{ɛ}(k)} + {B\; {\mu ( {k + 1} )}}}}{{n(k)} = {{g( {u(k)} )}^{\Omega_{l_{u},m_{u}}^{u}{(k)}}^{{\Omega_{l,m}{(k)}} + {{Cx}_{ɛ}{(k)}}}}}} & (3)\end{matrix}$

where Equation (3) may be obtained by taking the exponential of Equation(2) and by replacing ε(k) with Cx_(ε)(k). The system behavior may bedefined by the parameter vectors x₀, x_(u), and the ARMA model (A, B,C), and by the noise variance σ_(μ) ². That is, the volume may becomputable if the state, such as parameter vectors x₀ and x_(u), theARMA model (A,B,C) are known. Assuming u(k) is non-negative, then g(•)may vary depending on whether the control input is additive ormultiplicative:

${g( {u(k)} )} = \{ \begin{matrix}{\log ( {u(k)} )} & {{{{if}\mspace{14mu} u(k)} > {0\mspace{14mu} {and}\mspace{14mu} {is}\mspace{14mu} {multiplicative}}}\;} \\{{{sgn}( {u(k)} )}{\log ( {{{u(k)}} + 1} )}} & {{if}\mspace{14mu} {u(k)}\mspace{14mu} {is}\mspace{14mu} {additive}}\end{matrix} $

In some embodiments, one can define g(u(k)) to allow u(k) to benegative.

The model for the performance rate may be similar to that constructedfor the volume rates. The performance rate may be modeled withcontrol-signal-independent, control-signal-related, and/or errorcomponents. One may chose, for example, to model thecontrol-signal-independent and control-signal-related components withtruncated Fourier series with 24 and 168 hour periodic harmonics. Theerror component ε(k) may be modeled with an ARMA process or as a singleparameter in the system state x. For example, performance rate may bemodeled by the following logit-linear model:

logit(p(k)) = log (g(u(k)))Ω_(l_(u), m_(u))^(u)(k) + Ω_(l, m)(k) + ɛ(k)ɛ(k) ∼ ARMA(p, q)${p(k)} = \frac{{g( {u(k)} )}^{\Omega_{l_{u},m_{u}}^{u}{(k)}}^{{\Omega_{l,m}{(k)}} + {ɛ{(k)}}}}{1 + {{g( {u(k)} )}^{\Omega_{l_{u},m_{u}}^{u}{(k)}}^{{\Omega_{l,m}{(k)}} + {ɛ{(k)}}}}}$

FIG. 5 is a flow chart of an exemplary method for updating a model of anaspect of an advertisement. Consistent with certain embodiments of thepresent invention, an aspect of an advertisement may be modeled using amathematical model that has a control-signal dependant component, acontrol-signal independent component, and an error component. In steps520, 530, and 540, the three components are updated. In step 550, anupdated prediction model is generated based on the updated components.The updating steps 520, 530, and 540 may be performed separately ortogether as part of one or more operations and may be performed insequence or simultaneously. In some embodiments, for example, steps 520,530, and 540 may be performed together by using an adaptive estimationscheme or by updating all of the parameters of a particular type ofKalman filter, such as an unscented Kalman filter or an extended Kalmanfilter, where the parameters updated with the Kalman filter representall of the parameters of the control-signal-related component, thecontrol-signal-independent component, and/or the error component. Inother embodiments, one or more of the components are updated separatelyvia a different Kalman filter or by a different method. Other methods ofupdating components will be apparent to those skilled in the art fromconsideration of the specification and practice of the inventiondisclosed herein.

In some embodiments, in time period k (e.g., hour k), the volume (e.g.,n(k)) or performance (e.g., p(k)) and the control signal u(k) used forhours 1 to k may be known. Since the sourced volume data (e.g., clickvolume or conversion volume) in hour k may not be known for hours afterthe advertisement is displayed, one may interpret hour k as the lasthour for which there is sufficient volume or performance data. Forexample, after a certain duration of time, e.g., 1, 5, 10, 20, 24, 100,168, or 376 hours after an impression is sourced, a sufficientpercentage of the volume or performance data may be available. Asufficient percentage may be 50%, 80%, 90%, 95% or 99%, for example. Insome embodiments, an analysis of historical data for other similaradvertisements or campaigns may provide an estimate of what duration oftime is likely to be needed before a sufficient percentage of sourcedvolume data has arrived.

An extended Kalman filter may be used to estimate the2(1+l+m+l_(u)+m_(u)) component parameters and those of the error termε(k), which may be combined to produce:

${\begin{bmatrix}x_{0} \\x_{u} \\x_{ɛ}\end{bmatrix}( {k + 1} )} = {{{\begin{bmatrix}I & 0 & 0 \\0 & I & 0 \\0 & 0 & A\end{bmatrix}\begin{bmatrix}x_{0} \\x_{u} \\x_{ɛ}\end{bmatrix}}(k)} + {w(k)}}$log (n(k) + 1) = g(u(k))Ω_(l_(u), m_(u))^(u)(k) + Ω_(l, m)(k) + Cx_(ɛ)(k)

where the dimensions of the two identity matrices I may be(1+2(l+m))×(1+2(l+m)) and (1+2(l_(u)+m_(u)))×(1+2(l_(u)+m_(u))),respectively and A may be an s×s matrix. Moreover, the noise vector w(k)may be a zero-mean, Gaussian white noise process incorporating the whitenoise process driving the ε(k) ARMA process and artificial noise in x₀and x_(u). Furthermore, v(k) may be a zero-mean, Gaussian white noiseprocess describing the measurement uncertainty. In some embodiments:

Cov(w(k))=Q

Cov(v(k))=R

for all k, where Q and R are known covariance matrices.

In some embodiments, the measurement equation may be stated in terms oflog(n(k)+1) rather than log(n(k)) in order to avoid computational andnumerical issues, such as the undefined nature of log(0).

A linearized measurement signal may be used to allow use of a Kalmanfilter, such as an unscented or extended Kalman filter. Such ameasurement signal y(k) may be defined as follows:

y(k):=log(n(k)+1).

x may denote the state vector of the extended Kalman filter:

$x:={\begin{bmatrix}x_{0} \\x_{u} \\x_{ɛ}\end{bmatrix}.}$

F may represent the state transition matrix that relates x(k) and x(k+1)and may be defined as follows:

$F:={\begin{bmatrix}I & 0 & 0 \\0 & I & 0 \\0 & 0 & A\end{bmatrix}.}$

h(k,x(k),u(k)) may denote the nonlinear, time-varying measurementfunctional that relates the time index k, the state vector x(k) and thecontrol signal u(k) (but not measurement noise) to the measurement y(k).As such, it may follow that:

h(k,x(k),u(k)):=g(u(k))Ω_(l) _(u) _(,m) _(u) ^(u)(k)+Cx _(ε)(k).

As such, the state-space model may be written as:

x(k+1)=Fx(k)+w(k)y(k)=h(k,x(k),u(k))+v(k).

The partial derivative of h(k,x(k),u(k)) with respect to x may bedenoted H(k,x(k),u(k)) and may be given by the following row vector

$\begin{matrix}{{H( {k,{x(k)},{u(k)}} )}:=\frac{\partial{h( {k,{x(k)},{u(k)}} )}}{\partial x}} \\{= \lbrack {\frac{\partial{\Omega_{l,m}(k)}}{\partial x_{0}},{{g( {u(k)} )}\frac{\partial{\Omega_{l_{u},m_{u}}^{u}(k)}}{\partial x_{u}}},C} \rbrack}\end{matrix}$

where the partial derivatives of Ω_(l,m)(k) with respect to x₀ and Ω_(l)_(u) _(,m) _(u) ^(u)(k) with respect to x_(u) may be:

${\frac{\partial{\Omega_{l,m}(k)}}{\partial x_{0}} = \begin{bmatrix}1 \\{\sin ( {\frac{2\pi \; k}{24} + \phi_{{24\; {hr}},1}} )} \\{\beta_{{24\; {hr}},1}{\cos ( {\frac{2\pi \; k}{24} + \phi_{{24\; {hr}},1}} )}} \\\vdots \\{\sin ( {\frac{2\pi \; {lk}}{24} + \phi_{{24\; {hr}},l}} )} \\{\beta_{{24\; {hr}},l}{\cos ( {\frac{2\pi \; {lk}}{24} + \phi_{{24\; {hr}},l}} )}} \\{\sin ( {\frac{2\pi \; k}{168} + \phi_{{168\; {hr}},1}} )} \\{\beta_{{168\; {hr}},1}{\cos ( {\frac{2\pi \; k}{168} + \phi_{{168\; {hr}},1}} )}} \\\vdots \\{\sin ( {\frac{2\pi \; {mk}}{168} + \phi_{{168\; {hr}},m}} )} \\{\beta_{{168\; {hr}},m}{\cos ( {\frac{2\pi \; {mk}}{168} + \phi_{{168\; {hr}},m}} )}}\end{bmatrix}^{T}},{and}$$\frac{\partial{\Omega_{l_{u},m_{u}}^{u}(k)}}{\partial x_{u}} = \begin{bmatrix}1 \\{\sin ( {\frac{2\pi \; k}{24} + \phi_{{24\; {hr}},1}} )} \\{\gamma_{{24\; {hr}},1}{\cos ( {\frac{2\pi \; k}{24} + \phi_{{24\; {hr}},1}} )}} \\\vdots \\{\sin ( {\frac{2\pi \; l_{u}k}{24} + \phi_{{24\; {hr}},l}} )} \\{\gamma_{{24\; {hr}},l}{\cos ( {\frac{2\pi \; l_{u}k}{24} + \phi_{{24\; {hr}},l}} )}} \\{\sin ( {\frac{2\pi \; k}{168} + \phi_{{168\; {hr}},1}} )} \\{\gamma_{{168\; {hr}},1}{\cos ( {\frac{2\pi \; k}{168} + \phi_{{168\; {hr}},1}} )}} \\\vdots \\{\sin ( {\frac{2\pi \; m_{u}k}{168} + \phi_{{168\; {hr}},m}} )} \\{\gamma_{{168\; {hr}},m}{\cos ( {\frac{2\pi \; m_{u}k}{168} + \phi_{{168\; {hr}},m}} )}}\end{bmatrix}^{T}$

One may estimate x recursively from the measured volume data. Theestimate at time k may be denoted {circumflex over (x)} and may bedefined by

{circumflex over (x)}(k)=E[x(k)|y(0),y(1), . . . ,y(k)]

One may initialize (e.g., when k=0) the Kalman filter as follows:

${\hat{x}(0)} = \begin{bmatrix}{\log ( {{n(0)} + 1} )} \\0 \\\vdots \\0\end{bmatrix}$ P(0) = P₀(n(0))

where n(0) is the initial volume (at time k=0, e.g.) and P₀ may be aninitial estimate of the covariance of the state estimate.

At each subsequent time period (e.g., k=1, 2, . . . ), one may computethe state estimate propagation using the equation:

{circumflex over (x)} ⁻(k)=F{circumflex over (x)}(k−1),

compute the error covariance propagation using the equation:

P ⁻(k)=FP(k−1)F ^(T) +Q,

compute the Kalman gain matrix using the equation:

G(k)=P ⁻(k)H ^(T)(k,x(k),u(k))[H(k,x(k),u(k))P ⁻(k)H^(T)(k,x(k),u(k))+R] ⁻¹,

perform the measurement calculation using the equation:

y(k)=log(n(k)+1),

update the state estimate using the equation:

{circumflex over (x)}(k)={circumflex over (x)} ⁻+G(k)(y(k)−h(k,{circumflex over (x)} ⁻(k),u(k)),

and update the error covariance using the equation:

P(k)=(I−G(k)H(k,x(k),u(k)))P ⁻(k).

In order to update a performance model, a method similar to thatdescribed above for volume measurement may be used, except that theparameters that define p(k) may be updated based on received data, andthe measurement signal may be defined as:

${y(k)} = {{{logit}( {p(k)} )} = {{\log ( \frac{p(k)}{( {1 - {p(k)}} )} )} = {{{g( {u(k)} )}{\Omega_{l_{u},m_{u}}^{u}(k)}} + {\Omega_{l,m}(k)} + {{{Cx}_{ɛ}(k)}.}}}}$

Returning to FIG. 4, the updated model produced in step 420 is used topredict the aspect of the advertisement (step 430). In some embodiments,plant estimation module 310 performs step 420 and generates parametersthat define the model and sends these parameters to control module 320.Control module 320 may then use the parameters to predict the aspect ofthe advertisement. In some embodiments, additional processing (notdepicted in FIG. 4) may be useful for controlling or attempting tocontrol the aspect (e.g., attempting to control future volume orperformance). As such, the additional processing may include predictingthe effect that altering the control signal may have on the aspect and,based on that, choosing a control signal value that should result in adesired level for the aspect. For example, if the additional processingwas used to control conversion volume, and it was predicted that a valueof u_(desired) for control signal u should result in the desired (ornearest to the desired) conversion volume, then the control valueu_(desired) may be used to attempt to control the plant associated withthe advertisement.

In some embodiments, steps 410, 420, and 430 may deal with data relatedto groups of advertisements, such as campaigns. In such embodiments, thedata received in step 410 may relate to the group of advertisements; theupdated model produced in step 420 may model an aspect of the group ofadvertisements, such as volume or performance; and the aspect predictedin step 430 may be related to the group of advertisements. Similarly, insome embodiments, steps 510, 520, 530, 540, and 550 may deal with datarelated to groups of advertisements, such as campaigns. In suchembodiments, step 510 may include modeling an aspect of the group ofadvertisements and steps 520, 530, and 540 may include updating thecomponents of the model of the group of advertisements. Further, theaspect predicted in step 550 may be related to the group ofadvertisements. Whereas some of the discussion herein relates toindividual advertisements, it should be obvious to one skilled in theart how to adapt the methods, apparatuses, and system presented to beused with groups of advertisements.

In some embodiments, the steps depicted in FIGS. 4 and 5 are performedby one or more of the modules depicted in FIG. 3. In some embodiments,plant estimation module 310, control module 320, or plant module 330 mayindividually perform every step 410, 420, 430, 510, 520, 530, 540,and/or 550. In other embodiments, plant estimation module 310, controlmodule 320, or plant module 330 may perform fewer than all of step 410,420, 430, 510, 520, 530, 540, and 550. For example, plant estimationmodule 310 may perform steps 410, 420, 510, 520, 530, and 540 andcontrol module 320 may perform steps 430 and 550. In yet otherembodiments, one or more of steps 410, 420, 430, 510, 520, 530, 540,and/or 550 are performed by modules, apparatuses, or devices distinctfrom modules 310, 320, or 330.

Other embodiments of the invention will be apparent to those skilled inthe art from consideration of the specification and practice of theinvention disclosed herein. It is intended that the specification andexamples be considered as exemplary only, with a true scope and spiritof the invention being indicated by the following claims.

1. A method for analyzing advertisement-related data comprising:receiving data related to an aspect of an advertisement; modeling theaspect of the advertisement with a mathematical model, wherein themathematical model comprises a control-signal-related component, acontrol-signal-independent component, and an error component; updatingeach component of the mathematical model based on at least one of aparticular control signal, the received data, and a previous state of atleast one of the components to create an updated model; and predictingthe aspect of the advertisement using the updated model. 2-45.(canceled)